Photon Mean Free Paths, Scattering, and Ever-Increasing Telescope Resolution

Abstract

We revisit an old question: what are the effects of observing stratified atmospheres on scales below a photon mean free path λ? The mean free path of photons emerging from the solar photosphere and chromosphere is ≈ 10² km. Using current 1 m-class telescopes, λ is on the order of the angular resolution. But the Daniel K. Inoue Solar Telescope will have a diffraction limit of 0.020″ near the atmospheric cutoff at 310 nm, corresponding to 14 km at the solar surface. Even a small amount of scattering in the source function leads to physical smearing due to this solar “fog”, with effects similar to a degradation of the telescope point spread function. We discuss a unified picture that depends simply on the nature and amount of scattering in the source function. Scalings are derived from which the scattering in the solar atmosphere can be transcribed into an effective Strehl ratio, a quantity useful to observers. Observations in both permitted (e.g., Fe i 630.2 nm) and forbidden (Fe i 525.0 nm) lines will shed light on both instrumental performance as well as on small-scale structures in the solar atmosphere.

Appendices

Appendix A: Mean Free Paths and Scale Heights

Photons emerging from an optically thick atmosphere arise mostly from regions centered around optical depth τ=1. As Landi Degl’Innocenti (2013) has pointed out, in stratified layers this has the curious property that irrespective of the strength of the transition, the photons emerge from a region with an intrinsic thickness of ≈ 1 pressure scale height, when the opacity is proportional to pressure. This scale represents a basic limit below which one cannot say with certainty that a certain photon arose from or “forms at” a specific height. This statement applies to all features formed in optically thick stratified layers, in particular throughout the photosphere and the subsonic (non-spicular) components of the chromosphere.²

We can show this result formally. Let z measure the local height above some arbitrary point in the Sun, with the observer at ∞. The photosphere and low chromosphere comprise a partially ionized stratified layer with T≈6000 K, and a pressure scale height

$$ H = - \biggl(\frac{\mathrm{d}\ln p}{\mathrm{d}z} \biggr)^{-1} = \frac{kT}{\mu m_{\mathrm{H}} g} \approx130 \mbox{ km}. $$

(13)

Across the solar photosphere-chromosphere, temperature changes by less than a factor of two, whereas the pressure changes by orders of magnitude. Thus both gas density and pressure vary roughly as e^−z/H. We define the optical depth as usual,

$$ \tau(z) = - \int_\infty^z k(s) \,\mathrm{d}s, $$

(14)

where k(s)=λ(s)⁻¹ and λ(s) is the local mean free path of photons. We can also factor the opacity into k(s)=κ(s)ρ(s), where κ is the opacity in cm² g⁻¹, often a slowly varying quantity, and ρ(s) the mass density in g cm⁻³.

We define the region from which most photons emerge as between, for example, τ=1/e and τ=e. The geometrical thickness of this region is Δ=z ₁−z ₂, where

$$ \mathrm{e}^{-1} = -\int_\infty^{z_1} k(s) \,\mathrm{d}s, \qquad\mathrm{e}^{+1} = -\int_\infty^{z_2} k(s) \,\mathrm{d}s. $$

(15)

In the photosphere, opacity is dominated by the minority species H⁻ (e.g. Allen, 1973). In this case κ∝n _e, where n _e is the electron density, and in this case, n _e∝ρ since electrons come from singly ionized Fe, Si, etc. (There is also a temperature dependence through the Saha H, e and H⁻ equilibrium, which is important in deep photospheric layers, but ignored here for simplicity.) Above the photosphere, κ is dominated by lines and continua of neutrals and singly ionized ions (H⁻ being negligible at lower densities), and the opacity κ(s)≈constant (e.g. Vernazza, Avrett, and Loeser, 1981). We can write, roughly,

$$\kappa(s) = \kappa(0) \biggl[ \frac{\rho(s)}{\rho(0)} \biggr] ^m, $$

where m=1 (photosphere: H⁻) and m=0 (chromosphere). For a simply stratified atmosphere, we therefore have

$$ k(s) = k(0) \mathrm{e}^{-(m+1)s/H}. $$

(16)

Combining this with Equation (15), we have

$$ \mathrm{e}^{-1} = \frac{k(0) H}{m+1} \mathrm{e}^{-z_1(m+1)/H}, \qquad \mathrm{e}^{+1} = \frac{k(0) H}{m+1} \mathrm{e}^{-z_2(m+1)/H}. $$

(17)

The thickness of the region emitting most observed photons (i.e. those actually emerging from the atmosphere) is simply

$$ \Delta=(z_1-z_2) = 2H/(m+1). $$

(18)

Now, the mean free path of a photon where τ=1 is 1/k(τ=1) must be ≈ Δ: if it were larger (e.g., 10Δ), then this would mean that photons ten times deeper would escape, contrary to our assertion that they escape from τ=1; if smaller, the photons would not escape, thus

$$ \lambda\approx\Delta= 2H/(m+1). $$

(19)

(The factor of two arises simply because we adopted brightness factors e and 1/e.) The photon path length is on the same order as the pressure and density scale height in the photosphere (m=1) and lower chromosphere (m=0).

Appendix B: Surely the Chromosphere Is Not Hydrostatic?

When fluid motions are subsonic, the atmosphere cannot be far from a hydrostatic state. This is because sound or magnetosonic waves propagate changes in pressure to make the atmosphere approach a balance between pressure gradients and gravity. Much attention has been drawn to chromospheric features with highly supersonic motions. The classic semi-empirical, hydrostatic models of VAL (e.g. Vernazza, Avrett, and Loeser, 1981) have been much maligned, the chromosphere at the limb appears at a first glance to be extended much beyond the 1.5 Mm thickness of the model. So what is going on here?

The features moving supersonically include spicules and rapidly moving features seen on the solar disk. We can estimate what fraction of the Sun is covered at any given time by these rapid events using numbers recently published by Sekse and colleagues (Sekse, Rouppe van der Voort, and De Pontieu, 2012), revising earlier estimates by Judge and Carlsson (2010). The former authors find up to 2×10⁵ of these features on the Sun at any given time. Each is typically ℓ=2.5 Mm long and at most w=0.2 Mm across. Thus the area covered by these features is < wℓ=10⁵ Mm², of a total solar surface area of 6×10⁶ Mm². Thus by these data, at most just 1.7 % of the solar chromosphere has supersonic motions associated with it. This appears broadly consistent with the images shown by these authors, only a small fraction of the solar surface shows these features.

It has been claimed by some, however, that there is “no chromosphere” when there is no clear fibril structure. Given that many fibrils are supersonic, the claim might be extended to mean that there is no hydrostatic layer. It is difficult to assess what this claim might mean since the Sun cannot go immediately from photospheric pressures to coronal pressures with no intervening plasma pressure or magnetic stress – they differ by five (!) orders of magnitude. While claims and opinions may change, data do not. It is clear, and has been for some time, that if one observes narrow chromospheric line profiles on the solar disk, features with supersonic motions are quite rare. Using observations of C i emission lines from HRTS, for example, Dere, Bartoe, and Brueckner (1983) found that supersonic chromospheric “jets” are born at a rate of some 3×10⁻⁴ Mm⁻² s⁻¹ with a lifetime of about 40 s, and each has an area of some 1 Mm². Using these figures, we see that they cover only about 1 % of the total area of the chromosphere.

Therefore, we must recognize that the bulk of the chromosphere is approximately stratified in hydrostatic equilibrium. This is also consistent with early analyses of the solar flash spectrum (Athay, 1976), and the existence of 3 – 5 min oscillations throughout much of the chromosphere requires a hydrostatic stratification. As far as the chromosphere is concerned, the supersonic features are interesting, but fill only a tiny fraction of the entire chromosphere.